A correlation for the second interaction virial coefficients and enhancement factors for moist air

JOURNAL OF RESEARCH of National Bureau of Standards- A. Physics and Chemistry
Vol. 79A, No. 4, July-August 1975

A Correlation for the Second Interaction Virial Coefficients
and Enhancement Factors for Moist Air*

R. W. Hyland

Institute for Basic Standards, National Bureau of Standards, Washington, D.C. 20234

(March 28, 1975)

Experimental measurements <>f the enhancement factors for mixtures of water vapor and ( '.( )•_> -free
air have been made at —20, —10, and +70 °C. The results, coupled with previous experimental enhance-
ment data, have been used to calculate the second interaction virial coefficients, li mi , for water vapor
air mixtures from —50 to +90 °C. Within this temperature range, an error analysis shows that the
uncertainties in B nu - are between 6 and 10 percent. The calculated B llir values are used in deriving
enhancement factors at 10 °C intervals for — 50 < t < 90 °C, at varying pressure intervals from 0.25
to 100 bar. The associated uncertainties are shown as a function of pressure and temperature. The
enhancement factors are extrapolated to —80 °(!.

Key words: Enhancement factor; equation of state; interaction virial coefficients; moist air; saturated
air; second virial coefficients; virial coefficients.

1. Introduction

The objective of this paper is to take limited
experimental data on the saturated water vapor content
of C02-free air and develop a sound basis for predicting
this saturated content over wide ranges of temperature
and pressure. The approach is through the virial equa-
tion of state, and follows the methods outlined pre-
viously by Hyland and Wexler [1, 2], 1 where they
reported experimental results for the range
30 ^ t ^ 50 °C. In this paper the experimental range is
extended to include —20^ £^70 °C, and the results
used to obtain both the second interaction virial coeffi-
cients, Baw, for air-water vapor mixtures, and enhance-
ment factors, defined below, over the temperature
range -50 ^ t ^ 90 °C.

The experiments require the saturation of C0 2 -free
air at known conditions of pressure and temperature.
The water vapor is then separated from the air, and the
mole fractions of each component determined. The
enhancement factor for the particular pressure and
temperature of saturation may be expressed by

/=

x n P (l-x„)P

e,

e«

(1)

where /= the enhancement factor.
x„, *, r — the mole fractions of air and water vapor in
the saturated mixture.

*Thl8 work sponsored in part by ERDA, Sandia Laboratories, Primary Standards Labo-
ratory, Albuquerque, N. Mex. 87115.

1 Figures in brackets indicate literature references at the end of this paper.

P = total pressure above the surface of the

condensed phase (water or ice).

e, s = the pure phase saturation vapor pressure of

water substance at the temperature of

saturation.

x w P = the effective water vapor pressure at the

given pressure and temperature of saturation.

The determination of an empirical relationship for/
as a function of pressure and temperature is imprac-
tical because of the large number of experiments
involved. We therefore chose to use the theoretical
equation of [2], given as eq (A-4) in the appendix to
this paper. This equation yields enhancement factors,
providing all required virial coefficients are known with
sufficient accuracy. This was the case for all virial
coefficients within our range of interest except B (nr .
Our approach was to perform a limited number of
enhancement factor measurements, and, by inverting
eq (A— 4) into a quadratic for Z?, /Ir , to calculate B (nc at
each experimental point. The values obtained at each
temperature were averaged, and the temperature
dependence of the averages estimated by least-squares
fitting them with the Lennard-Jones (12-6) potential
function, resulting in the values found in table 4 of this
paper. All of the information necessary for using
eq (A— 4) was then in hand, and the enhancement
factors given in table 8 were obtained.

2. Experimental Method and Results at —20,
-10, and+70°C

Commercially supplied, compressed air, with
impurity characteristics as described in [1], is passed

551

through a unit which lowers the C0 2 content to a level
near 2 ppm. The air is then saturated by passage over
the surface of either water or ice. Pressure and tem-
perature within the saturator are monitored. The water
vapor is subsequently removed from the air stream, and
the mass of water vapor and associated dry air deter-
mined. One obtains the water vapor mole fraction from
the mass ratio, and the pure phase saturation vapor
pressure from the saturator temperature. Combining
these with the saturator pressure in eq (1) determines
the enhancement factor at the saturator P, T condition.
Several such experiments are done along an iso-
therm. Since Baw is a function of temperature only,
each experiment should yield the same value, providing
a check on whether or not the apparatus is performing
properly, and whether the equation relating B aw to /
(the inverse of eq (A— 4) in this paper) is valid at the
given conditions.

The details of the methods for experiments above
°C, including our new point at +70 °C, are found in
[1, 2]. The 70 °C data and results are in table 2 of this
paper.

The saturating apparatus of [1, 2] is unsuitable for
use at temperatures below °C. Instead, for our points
at —20 and —10 °C, the low-frost point generator of
Greenspan [3] was used. Air samples were prepared
by first evacuating containers, then back-filling them
through a C0 2 -removal unit in series with a coil
immersed in a bath at roughly —78 °C. The air pressure
during filling increased monotonically from about 1 to
125 bar. The process provided containers of dry,
I0W-CO2 content (2-5 ppm) air. In the course of an
experiment, the air mass passing through the saturator
was found by weighing the air containers before and
after the experiment on a modified, higher-capacity
version of a balance described by Russell in [4]. The
water mass was determined with the usual gravimetric
drying train [5].

The experimental results at —10 and —20 °C are
given in table 2, while table 3 summarizes the results
obtained previously at 30, 40, and 50 G C.

All experimental results are referred to the Inter-
national Practical Temperature Scale of 1948 (6). In
table 4, above —20 °C, it makes no difference whether
the given temperatures are considered to be on
IPTS-48 or IPTS-68 (7). At -20, -30, and -40 °C,
B (l w becomes more negative by 0.01 cm 3 /mol, and at
—50 °C, 0.02 cm 3 /mol, if one considers the given
values to be on the 1948 scale and converts to the 1968
scale. In table 8, to the number of reported places, the
temperatures can be considered to represent either
temperature scale.

3. Data Reduction

We first derive values of B aw from enhancement data.
The necessary equations have been discussed in
refs. [1, 2]. However, the derivation of eq (12) of [2]
is incorrect as presented, and the proper derivation
is outlined in the appendix to this paper. Also sum-
marized in the appendix are equations in which no
changes have been made. Presented here are equations

and constants which differ from those used in [1, 2].
These differences are associated with the wider
temperature ranges considered. Also, the molecular
weight of air containing 2-5 ppm CO2 was recalcu-
lated, based on the proposal by Harrison [8] that the
sum of the C0 2 and oxygen volume percentage is
constant at 20.979. The molecular weight ratio of
water to air becomes 0.622062, differing slightly from
the value used in [1, 2].

3.1. Specific Volume of Hexagonal Ice I

The specific volume of hexagonal ice I, used in the
calculation of B (llv from experimental enhancement
data at temperatures below °C, was formulated by
first considering the specific volume data of Ginnings
and Corruccini [9] , along with a correlation of data on
the linear expansion of ice from Jukob and Erk [10],
Powell [11], Butkovich [12], Dantl [13], and LaPlaca
and Post [14], resulting in

Vp a , T = 1.074351- 0.867004- 10 4 r+0.656565-10 6 T 2
-0.430316-10" 9 T» (2)

where Vp a ,T is the specific volume of ice at one standard
atmosphere and at absolute temperature T.

Equation (2) is used in our correlation of Leadbetter's
values [15] of the adiabatic compressibility of hexagonal
ice I, to yield

V Pt T = Vp atT [l-(8.875 + QM65T)

(P-1.01325)-10- 6 ] (3)

where the pressure unit is bars. 2

3.2. Expressions for Baa, Caaa, and Caaw

Equations (4), (5), and (6), below, were obtained
by fitting, using the method of least squares, a wider
range of data of Sengers et al. [16], Hilsenrath et al.
[17], and Mason and Monchick [18] than was done
in [2] . Polynomial equations were used for the fits. The
data fitted here covers the nominal temperature range
from —50 to +110 °C. The standard deviations of the
fits do not, of course, reflect the overall uncertainties
in the parameters.

fl„«(0 =-13. 5110 + 0.24311* -0.10846-10" 2 ; 2

+0.42504-10-V- 0.81851 -lO^ 4 cm 3 /mol;
<r m = 0.023 (4)

Caaa (t ) = 1314.2 - 0.89988; - 0.30474- 10 ~H 2
+0.42015-10-V-0.40869-10- 6 ; 4
+0.43810-10- 8 ^-0.20677-10- 8 f 6 cm 6 /mol 2 ;
o- m = 0.157 (5)

Caaw(t) =860.82 -2.4454* + 0.94106-10 2 t 2
+0. 14909- 10" 4 ^- 0.59389- 10- V
+0.30265-10- 8 * 5 cm 6 /mol 2 ; cr fit = 0.353 (6)

The temperature, t, is degrees Celsius.

'■ 1 bar= 10 5 pascals

552

3.3. Expressions for the Henry's Law "Constants"

The correlation of the Henry's "constant", k, for air,
from to 100 °C, is based in part on the "constants"
for nitrogen, as was done for the to 50 °C range of [2].

For atmospheric nitrogen, correlations were obtained
from the data of [19, 20, 21], and, for nitrogen at pres-
sures of 50 and 100 atmospheres, from the data of [22J
as reported in Dorsey [21], and from [23].

The 50- and 100-atmosphere air curves were obtained
by requiring the same percentage differences between
the 1- and 50-atmosphere curves and the 50- and 100-
atmosphere curves as the corresponding differences
for nitrogen.

The results, using bars as the pressure units, were
fitted to the polynomial

106 k = j? QiV (7)

»' =

where rc = 4 or 5, yielding the coefficients in table 1.
The units of k are mole fraction/bar.

There seems to be no high pressure data for
solubility of air in ice. At all temperatures less than
°C, the Henry's "constant" for water at °C and 1 bar
is used. The resulting systematic error estimate, given
as 10 percent in [2], has been increased to 20 percent,
introducing a maximum uncertainty on the order of
0.2 percent into the calculated value of B< m -

TABLE 1. Coefficients to equation (7)

Pressure, bars

i

1

50

100

1
2
3
4
5

23.5199

-0.60277
.11518-10'

-.12556-10 ;{
.74217-10 8
.17765-10 H

21.6663
-0.54460
.93580 -lO" 2
-.75473-10 4
.24146 • 10 "

19.7652
-0.47424
.81569-10 *
.67051 -10-*
.22157-10 -«

3.4. Saturation Vapor Pressure Over Ice

The expression for the saturation vapor pressure of
ice is that given by Goff, in [24J. At -10 and -20 °C,
the estimated systematic uncertainty is 0.1 percent in
the predicted vapor pressure.

GofFs equation is

log,o (e,) =-9.096936 [273.16/(7-1)]
-3.56654 log 10 (273.16/7 7 )
+ 0.876817 (1-77273.16) +0.786118 (8)

where e\ = saturation vapor pressure of pure ice,
millibar
T= absolute temperature, kelvins.

This equation does not
at the triple point as that

jive the identical value of e s
of water used above °C (see

[25], but the difference, 1 X 10
the context of this paper.

mbar, is nej

digibl

e in

3.5. Theoretical Smoothing Function for Baw

In order to obtain an equation for the interpolation
and extrapolation of B au versus t, the Lennard-Jones
(12-6) potential was fitted to the mean, experimentally
based values of Saw found in tables 2 and 3, yielding the
calculated values of table 4. The potential parameters
are b = 51.994 cm 3 /mol, and e/A= 143.513 K. The data
are not randomly distributed about the predicted curve;
however, within the error limits assigned in section 5.2,
this is not significant.

The analysis of Hanley and Klein [26] shows that,
given a LJ (12-6) fit to data within the region where the
reduced temperature T * is between approximately 2.0
and 10.0, any two- or three-parameter function can be
fitted equally well to the data. Furthermore, in this
situation, extrapolation much beyond the experimental

Table 2. Summary of experimental conditions and results

Saturation

Saturation

Total

Interaction

Normalized b

temp.

vap. press.

press.

Mole fraction

Enhancement

virial coeff.

inter, vir.

°C

mb

bar

water vapor

factor a

cm'Vmol

coefficient

cm'Vmol

t

e.s

P

x w

/

Baw

Baw

70.0067

311.7533

24.0502

0.013640

1.0523

-16.59

-16.61

70.0007

311.6726

29.5089

.011258

1.0659

-18.25

-18.25

69.9952

311.5984

32.8965

.010172

1.0739

-17.84

-17.83

69.9991

311.6510

34.9783

.0096093

1.0785

-18.18

Mea
std. dev. of men

-18.18

n « -17.72
n = 0.38

-10.0043

2.595475

19.9975

.00014031

1.0857

-40.47

-40.46

-10.0000

2.596456

29.0621

.00010028

1.1268

-41.17

Mec
std. dev. of met

-41.17

in = -40.82
m = 0.35

-20.0125

1.03023

19.9953

.000056020

1.0873

-43.29

-43.29

*f=x w P/e s .

b By noting the slope of Baw versus t, one adjusts the values of the previous column to correspond to exactly 70,

-10, or -20 °C.

553

region, particularly toward lower temperatures, may
be hazardous.

For our experiments, 1.76 < T * < 2.40, with two of
the six points being below jT*= 2.0. We verified that
(9-6) and (24-6) potential fits produced, to well
within experimental accuracies, the same results as the
(12-6) fit from at least —50 to +90 °C, then dropped
them from further consideration. The extrapolations
probably could be extended upwards by another 20 or
30 °C with little change in the accuracy. However,
because of the potential hazards of extrapolating down-
wards, we feel that assigning errors below —50 °C
would be meaningless.

Table 3. Mean values and uncertainties of the experimental NBS
virial coefficients at 30, 40, and 50 °C [2]

t

B\? x l

CT b

Systematic error

3cr+ syst.

°c

cm 3 /mol

Percent

Percent

Percent

30

-29.2

0.6

4

5.8

40

-26.3

.7

4

6.1

50

-23.6

1.4

6

10.2

a From [2].

b cr= standard deviation of the mean value of Baw

Table 4. Results of fits to B aw a

t

Baw, from LJU2-6)
cm 3 /mol

Baw, from eq (9)
cnr'Vmol

°c

-50

-58.0

-58.0

-40

-53.0

-52.9

-30

-48.4

-48.4

-20

-44.2

-44.2

-10

-40.4

-40.5

-37.0

-37.0

+ 10

-33.8

-33.8

20

-30.9

-30.9

30

-28.2

-28.2

40

-25.7

-25.6

50

-23.3

-23.3

60

-21.2

-21.2

70

-19.2

-19.2

80

-17.3

-17.3

90

-15.5

-15.5

a From LJ (12-6) fit to data of tables 2 and 3, b =
and eM=143.513 K.

51.9941 cnr'Vmol

In most cases it would be easier to use a polynomial
representation of B aw versus t, rather than to inter-
polate tables of Lennard-Jones values. Accordingly,
the values predicted from the Lennard-Jones fit
(column 2 of table 4), using the method of least squares,
were fitted to a polynomial function of Celsius tem-
perature, resulting in the equation

-B au = 36.98928 -0.331705* + 0.139035 • lO" 2 ^
-0.574154 • 10^ + 0.326513 • 10 7 £ 4
-0.142805 • 10" V cm 3 /mol, (9)

valid for — 50 < * < 90 °C. The values predicted by
eq (9) are given in the final column of table 4.

The maximum difference between the rounded

values of eq (9) and the Lennard-Jones values, is
seen to be 0.1 cm 3 /mol. Equation (9) supercedes eq
(37) of [2].

4. Uncertainties in Baw

In this section are presented, first, a discussion of
the uncertainties in the experimentally-based values
of Baw, and second, a discussion of the uncertainties
in the predicted values.

4.1. Uncertainties in the Experimentally Derived
Values

a. Contribution from Measured Quantities

At 70 °C, the systematic uncertainties from the
measured quantities may all be placed into the
measured enhancement factor, as was done in [2].
The systematic uncertainty in the measured enhance-
ment factor is 0.07 percent just as it was for the 30,
40, and 50 °C values of [1]. This includes all but
inconsequential effects from pressure, temperature
(and through it, the saturation vapor pressure), and
mole fraction of water vapor (as determined from the
measured mixed ratio), and contributes an estimated
2 percent systematic uncertainty to B a w>

The standard deviation of the mean value is used as
the measurement of the random uncertainty of the
70 °C value of B (m . As seen in table 2, this amounts to
2 percent. The first of the tabulated 70 °C results
appears low relative to the others; if it were dropped
from consideration, the standard deviation of the mean
would decrease to 0.7 percent. However, there seems
to be no valid reason for rejecting the point. (Note that,
in table 7, the overall uncertainty would become 7
percent instead of 12 percent.)

The remaining discussion in this section applies to
the —10 and —20 °C experiments. The standard
deviations for the pressure and temperature measure-
ments are the standard deviations of the means
obtained during the experiments, while the standard
deviations of the mixing ratios are based on the
standard deviations of the weighings of the water
absorption tubes and of the air containers.

The systematic uncertainty in the temperature
measurements is the sum of estimated systematic
uncertainties in the instrumentation, in the ice point
resistance of the resistance thermometer, and in the
resistance versus temperature curves used to represent
the thermometer calibration data, plus an allowance of
0.005 °C for the possible deviation of the measured
temperature from the temperature of saturation.

The systematic pressure uncertainty is primarily the
sum of estimated systematic uncertainties in the curve
used to represent the calibration of the manometer,
in the manometer zero reading, plus small contributions
from other corrections associated with the manometer.

The assigned systematic errors in the mixing ratio
account for the possibility of water passing unabsorbed
through the U-tubes [5], plus small contributions from
the bouyancy corrections applied to the air containers
during weighing and the effects of different starting
and finishing pressures in the high-pressure lines.

554

Table 5 summarizes the experimental error sources
and magnitudes, and the errors they induce in Baw,
for the experiments at — 10 and —20 °C.

b. Contributions from Calculated Quantities

The error contributions from the various calculated
quantities were obtained by changing the quantity by
the amount of its estimated error, recalculating B< lw ,
and comparing with the originally calculated value.

Sengers et al. [16| estimate the 3cr error in Baa to be
1.2 cm 3 /mol. Uncertainties in B W w, C a aa, and C www ,
were estimated by comparing available data, as de-
scribed in [2 1, or, in the cases of C (UIW and C a ww,

arbitrarily selected to indicate possible error magni-
tudes. The intent was to be conservative, but that
cannot be guaranteed. The estimated uncertainties
associated with the various virial coefficients are
summarized in table 6, along with the systematic errors
they induce in B a w Table 6 similarly lists the effects
of the uncertainties in the Henry's and gas constants,
and in the saturation vapor pressure. Below °C, we
indicate separately the error arising from use of eq (8).
Above °C, the 60 ppm error associated with the vapor
pressure equation [25 1 was lumped into the experi-
mental vapor pressure error. The final column of table
6 combines all contributions by quadrature, that is,
the square root of the sums of the squares. This is

Table 5. Experimental errors at —Id and-20°C

Nominal run

Estimated systematic

errors

3o--Random errors

Parameter

conditions

Parameter

Induced B„ u error

Parameter

Induced B

aw error

uncertainty

uncertainty

t(°C)

P(bar)

cm 3 /mol

Percent

cm 3 /mol

Percent

-10

20

t

0.017 °C

0.81

2.00

0.0003 °C

0.01

0.03

P

.0172 bar

.43

1.06

.00041 bar

.10

.25

r ;i

.07%

.38

.94

.11%

.59

1.46

2-1.62

4.00

Quad" = 0.60

1.48

-10

30

t

().017°C

0.55

1.33

0.0003 °C

0.03

0.07

p

.0159 bar

.18

.44

.0062 bar

.07

.17

r

.09%

.30

.72

.10%

.43

1.04

2=1-03

2.49

Quad = 0.44

1.06

-20

20

t

0.017 °C

0.82

1.89

0.0015 °C

0.07

0.1

P

.0172 bar

.41

.95

.0041 bar

.10

.23

r

.14%

.72

1.66

.10%

.51

1.18

1

4.50

Quad =0.52
1

1.21

n r— Mass water/mass air, g/g

b Combined by quadratures, i.e., the square root of the sum of the squares.

FABLE 6. Systematic uncertainties in Baw from calculated parameters

Press.

Source of error

Quadrature

Temp.

B„a

B wu .

C< aaa C www *~* aaw *■* airir

Other

e t

bar

Estimated error in source, percent

( a )

6

15 46 50 50

( b )

-

°C

Estimated error in B n w> percent

70

24.1

3.26

1.54

0.44 0.36 1.27 2.25

0.14

-

4.47

Estimated error in source, percent

( a )

100

20 100 150 150

( b )

0.1

Estimated error in Baw, percent

-10
-10
-20

20.0
29.1
20.0

1.34
1.42
1.32

0.31
.43
.20

0.41 0.00 3.11 0.19
.29 .00 2.22 .18
.29 .00 2.22 .09

0.16
.16
.16

1.34
0.88
1.19

3.69
2.84
2.87

' Estimated 3cr error is 1.2 cm 3 /moL

1 Quadrature of errors contributed by Henry's law, gas constant, and ignoring

the correction to the law of ideal solutions.

555

justified on the basis that it is unlikely that all errors
will contribute, in the same sign sense, to the un-
certainty in B„w

Table 7 summarizes the systematic and random
errors in the experimentally derived values of Baw> The
estimated contributions from the experimental and
calculated parameter systematic sources are shown
separately, then combined into a total systematic
uncertainty in column 6. The estimated overall un-
certainty at 70 °C is 12 percent. At — 10 °C, both a
calculated and observed estimate of the experimental
random errors are available, leading to two estimates
of the total error which are in good agreement. The
larger estimate is 7 percent. At —20 °C, the total un-
certainty is 6 percent.

signed. This is larger than the 7 percent estimate
given in table 7, but since the errors associated with
various other virial coefficients are only crude es-
timates in this region it is felt that the additional 3
percent on the error band is warranted. For ^ t < 45
°C, a 6 percent error band is assigned, commensurate
with the estimated uncertainties of table 3. Above
45 °C, the error band is again increased to 10 percent.
All experimental points lie within the error band of
the predicted values, and vice versa. If either or both
experimental end points (—20 or 70 °C) are omitted,
the resultant LJ (12-6) fit still lies within the assigned
bands over the entire range of interest.

5. Enhancement Factors

4.2. Uncertainties in the Predicted Values of B«, r

The maximum error bands placed on the predicted
values of table 4 reflect the maximum estimated un-
certainties in the experimental quantities. For —50^
t <0 °C, error bands of ±10 percent have been as-

The enhancement factor is defined by eq (1).

To obtain the enhancement factor at a given (P,
T) condition, one replaces the / in eq (A-4) of the
appendix by eq (1), solves by iteration for x a , then
converts back to / through eq (1).

Table 8 gives the results of such calculations. The

Table 7. Summary of errors in experimentally derived values o/B aw

Temp.

Error source

°C

Experimental
random, 3& A

Experimental
systematic

Calc. parameters
systematic

Quadrature
of systematics

Overall
quad. syst. 4-3(7"

Error in B (lu , percent

70
-10 b
-10 c
-20

6.42
1.26
2.57
1.21

2.38
3.24
3.24
4.49

4.47
3.26
3.26
2.87

5.06
4.59
4.59
5.33

11.5
5.9

7.2
6.5

a <x = Standard deviation of mean value.

b Based on means of the calculated uncertainties at 20 and 29 bar, tables 5 and 6.

c Random from the observed cf of table 2.

Table 8. Predicted enhancement factors

Total

t

°C

pressure

bars

-80 a

-70

-60

-50

-40

-30

-20

-10

0.25

1.0020

1.0018

1.0016

1.0014

1.0013

1.0012

1.0012

1.0012

.50

1.0041

1.0036

1.0032

1.0029

1.0026

1.0024

1.0022

1.0022

1.00

1.0082

1.0072

1.0064

1.0058

1.0052

1.0047

1.0044

1.0041

1.50

1.0123

1.0109

1.0097

1.0086

1.0078

1.0071

1.0065

1.0060

2.00

1.0165

1.0145

1.0129

1.0116

1.0104

1.0094

1.0086

1.0080

2.50

1.0207

1.0182

1.0162

1.0145

1.0130

1.0118

1.0108

1.0099

3.00

1.0249

1.0219

1.0195

1.0174

1.0156

1.0141

1.0129

1.0119

3.50

1.0291

1.0256

1.0228

1.0203

1.0183

1.0165

1.0150

1.0138

4.00

1.0334

1.0294

1.0261

1.0233

1.0209

1.0189

1.0172

1.0158

4.50

1.0377

1.0332

1.0294

1.0262

1.0236

1.0213

1.0194

1.0177

5.00

1.0420

1.0369

1.0327

1.0292

1.0262

1.0237

1.0215

1.0197

10.00

1.0865

1.0758

1.0669

1.0595

1.0533

1.0480

1.0435

1.0396

20.00

1.184

1.160

1.140

1.124

1.110

1.099

1.089

1.080

30.00

1.293

1.252

1.220

1.193

1.171

1.153

1.138

1.124

40.00

1.417

1.356

1.307

1.269

1.237

1.211

1.189

1.170

50.00

1.556

1.470

1.403

1.350

1.307

1.272

1.243

1.218

60.00

1.71

1.60

1.51

1.44

1.38

1.337

1.300

1.268

70.00

1.89

1.74

1.62

1.53

1.46

1.41

1.360

1.321

80.00

2.10

1.90

1.75

1.64

1.55

1.48

1.42

1.377

90.00

2.33

2.08

1.89

1.75

1.64

1.56

1.49

1.44

100.00

2.59

2.27

2.05

1.88

1.75

1.64

1.56

1.50

See footnote at end of table.

556

Table 8.
Predicted enhancement factors — Continued

Total

°C

bars

Ice

Water

0.25

1.00132

1.00131

.50

1.00221

1.00217

1.00

1.0040

1.0039

1.50

1.0057

1.0056

2.00

1.0075

1.0074

2*50

1.0093

1.0091

3.00

1.0111

1.0108

3.50

1.0129

1.0126

4.00

1.0146

1.0144

4.50

1.0164

1.0161

5.00

1.0182

1.0179

10.00

1.0364

1.0356

20.00

1.074

1.072

30.00

1.113

1.111

40.00

1.154

1.151

50.00

1.197

1.193

60.00

1.242

1.237

70.00

1.289

1.282

80.00

1.338

1.330

90.00

1.389

1.381

100.00

1.44

1.43

Table 8. Predicted enhancement factors— Continued

Total

t°C

bars

10

20

30

40

50

60

70

80

90

0.25

1.00148

1.00173

1.00202

1.00223

1.00211

1.00111

.50

1.00229

1.00251

1.00284

1.00323

1.00358

1.00362

1.00288

1.00051

1.00

1.00388

1.00400

1.00426

1.00467

1.00519

1.00571

1.00599

1.00564

1.00394

1.50

1.0055

1.00547

1.00564

1.00599

1.00651

1.00713

1.00772

1.00801

1.00754

2.00

1.0071

1.0069

1.00701

1.00728

1.00775

1.00839

1.00910

1.00968

1.00980

2.50

* 1.0087

1.0084

1 .0084

1.0086

1.0090

1.00959

1.01034

1.01108

1.01154

3.00

1.0103

1.0099

1.0097

1.0098

1.0102

1.0108

1.01151

1.01234

1.01300

3.50

1.0119

1.0114

1.0111

1.0111

1.0114

1.0119

1.0125

1.01351

1.01432

4.00

1.0135

1.0128

1.0125

1.0124

1.0126

1.0130

1.0138

1.0146

1.01553

4.50

1.0151

1.0143

1.0138

1.0136

1.0138

1.0142

1.0149

1.0157

1.0167

5.00

1.0167

1.0158

1.0152

1.0149

1.0150

1.0153

1.0159

1.0168

1.0178

10.00

1.0330

1.0308

1.0290

1.0277

1.0269

1.0265

1.0266

1.0271

1.0280

20.00

1.066

1.0615

1.0573

1.0539

1.0512

1.0493

1.0480

1.0474

1.0474

30.00

1.101

1.093

1.087

1.081

1.076

1.073

1.0698

1.0680

1.0670

40.00

1.138

1.126

1.117

1.109

1.102

1.096

1.092

1.0890

1.0869

50.00

1.175

1.161

1.148

1.137

1.128

1.121

1.115

1.111

1.107

60.00

1.215

1.196

1.180

1.167

1.155

1.146

1.139

1.133

1.128

70.00

1.256

1.233

1.213

1.197

1.183

1.172

1.163

1.155

1.149

80.00

1.298

1.271

1.248

1.228

1.212

1.198

1.187

1.178

1.171

90.00

1.343

1.311

1.284

1.261

1.242

1.226

1.212

1.202

1.193

100.00

1.389

1.352

1.320

1.294

1.272

1.254

1.238

1.226

1.216

a Values below
can be assigned.

-50

°C are based on a questionable extrapolation of B nu . They are given as a matter of interest, but no error bands

lowest tabulated pressures at 70, 80, and 90 °C are
exceeded by the saturation vapor pressure, precluding
the existence of a saturation equilibrium condition.
The enhancement factors calculated under those
conditions are meaningless in the context of this
paper, and have been omitted.

In the error discussion to follow, no attempt was
made to assign uncertainties below —50 °C, for reasons
discussed in section 3.

However, as a matter of interest, table 8 has been
extrapolated to —80 °C. The necessary B (lw extrapola-

tion used the Lennard-Jones (12-6) relationship, not
eq (9).

For those interested in interpolation, note that, at
low pressures, the enhancement factors increase
rapidly as the pressure increases from the value of
the saturation vapor pressure. The slope then suddenly
decreases, the curve becoming more linear. The loca-
tion of the slope change along the pressure axis, no
greater than about 3 bar for the temperatures con-
sidered here, is a function of temperature.

557

578-011 0-75-2

6. Uncertainties in the Predicted Enhancement
Ratios

In section 4, the contributions to the B aw uncer-
tainty from various calculated quantities (virial
coefficients, saturation vapor pressure, etc.) were
outlined. When the uncertainty in B a w is considered
as one computes the uncertainties in the predicted
enhancement factors, the contributions from the re-
maining parameters are directly included, as part of
the Baw error. Therefore, the only quantity of concern
in calculating the uncertainty in the predicted en-
hancement factors of table 8 is the interaction coeffi-
cient Baw For table generation, pressure and tem-
perature are considered exact. At —50 °C and above
the significant figures are given to the place which is
affected by the maximum uncertainty (table 9) by no
more than five units, plus one more place.

Table 9 outlines the estimated uncertainty in the
predicted enhancement factors as a function of
pressure and temperature. Linear interpolation of the
errors, between adjacent tabulated pressure along
the isotherm, allows error estimates to within 0.05
percent.

It is important to reemphasize that, below —50 °C,
the error bands are considered unknown and, at
present, unobtainable. Research at temperatures
below —20 °C will be needed for further B (lw extrapo-
lation and bracketing of the enhancement factors.

Table 9. Estimated maximum percentage uncertainties in
calculated f a

Table 10. Comparison of {-values with Webster

p

Temperature, °C

bar

90

70

50

30

+ 10

-10

-30

-50

1.00

0.001

0.01

0.01

0.01

0.03

0.04

0.05

0.06

2

.01

.02

.03

.03

.06

.07

.10

.13

3

.02

.03

.05

.04

.09

.11

.14

.19

4

.03

.05

.06

.05

.11

.15

.19

.25

5

.04

.06

.08

.07

.14

.18

.24

.32

10

.09

.13

.17

.13

.29

.37

.48

.64

20

.20

.26

.35

.27

.58

.75

.98

1.30

40

.41

.54

.70

.54

1.2

1.5

2.0

2.7

60

.62

.82

1.1

.82

1.8

2.3

3.1

4.2

80

.83

1.1

1.4

1.1

2.4

3.2

4.2

5.8

100

1.1

1.4

1.8

1.4

3.1

4.1

5.4

7.5

a No errors are assigned to values below —50 °C, nor should any
attempt be made to do so.

7. Comparison With Other Work

Comparisons with other virial coefficient work may
be found in [2].

The work of Goff [27] and Webster [28] are the two
extensive experimentally based works with which we
will compare our air-water vapor enhancement
factors.

Table 10 compares selected high pressure data with
Webster, while table 11 compares selected low pres-
sure data with Goff. The disagreement with the former
is within a factor of two of the NBS estimated maxi-
mum uncertainty except at —20 °C and 50 bar. How-

p

bar

Temperature, °C

-20

+ 50

50

Webster

1.36
1.243

9.4

1.75
1.563

12.0

1.24
1.193

3.9

1.50
1.433

4.7

1 14

NBS

1 128

Difference, percent

1.1

100

Webster

NBS

1.28

1 272

Difference, percent

0.6

Table 11. Comparison of {-values with Goff

p

Temperature

, °c

bar

60

20

-20

-50

0.25

Goff

1.0011

1.0036

1.0060
1.0057

1.0018
1.0017

1.0027
1.0025

1.0045
1.0040

1.0014
1.0013

1.0024
1.0022

1.0044
1.0040

1.0013
1.0012

1.0024
1.0022

1.0048
1.0044

1.0015

NBS

1.0014

0.50

Goff

1.0030

NBS

1.0029

1.00

Goff

NBS

1.0060
1.0058

ever, the discrepancies will increase as the tempera-
ture decreases. The disagreement between NBS and
Goff is within the NBS estimated uncertainty in all
cases, ranging from 0.01 percent at 0.25 bar to 0.05
percent at 1 bar.

8. Summary

New experimental enhancement factor data at
— 20, —10, and +70 °C, coupled with previously re-
ported values at 30, 40, and 50 °C, have been used to
obtain the air-water vapor interaction virial coefficient,
Baw, at those temperatures. The B a w values have in
turn been fitted by a Lennard-Jones (12-6) potential,
to permit extrapolation and interpolation. An empirical
equation for Baw versus temperature is also given.
Either equation predicts values considered uncertain
by no more than 10 percent, for — 50^ £^ °C and
for 45 ^ ^ 90 °C. For ^ t ^ 45 °C, the uncertainty
is 6 percent. Using these B aw values, enhancement
factors have been derived for air over ice from —50
to 0°C, and for air over water from to +90 °C, at
10 °C intervals, and at varying pressure intervals from
0.25 to 100 bar. The associated uncertainties vary
according to condition, and are best seen in table 9.
Extrapolations of the enhancement factor, to —80 °C
for air over ice, are presented, but no uncertainties are
assigned.

We thank the members of the Office of Weights and
Measures of the NBS, and Harry Johnson in par-
ticular, for their patience and assistance during our
weighings of the air containers on the Russell balance.

558

9. Appendix

The derivation of eq (12) of [2], which refers only
to the gas phase, begins with

Pi

dGj

(A-l)

where jjl and n are the chemical potential and number
of moles of water vapor, and G the Gibbs function,
associated either with the mixture (i = m) or pure
water vapor (i = w).

Equation (A-2) evaluates the Gibbs function
difference between the saturated equilibrium state
at pressure P and a reference state at pressure P () .

Q (T,P,n„,n,) - Q (7\ P , n,„ m) = \ VidP (A-2)

where n (l is the number of moles of air and V\ the
volume of the gas under consideration at temperature
T. The volumes V\ are obtained from the appropriate
virial equation of state. (Note, however, that the
magnitudes of n\ differ in the expressions for V m and
V w ).

The right side of (A— 2) is evaluated for the mixture
between P and the saturated equilibrium pressure P,
and for pure water vapor between P and the saturation
vapor pressure $. Equation (A-l) yields the corres-
ponding chemical potential differences. By subtracting
the results of these operations and rearranging, one
obtains the chemical potential difference A/x of the
water vapor between the equilibrium conditions at
pressure P in the mixture and e s in the pure phase.

Let P be low enough so that both the mixture
and pure water vapor can be considered as ideal gases
in the reference state. The term /x m (7\ P , rut , n m )
appearing in the expression for A/x can be written as

fi m (T, P , n a , n in ) = iXm(T, Po, 0, n m ) + RT\n x w (A-3)

where x w is the mole fraction of water vapor. Then,
since in the ideal gas limit /jl,,i(T, Po, 0, n,„) cancels
a similar term, — jul u (T, P , 0, n w ), and since all non-
ideal contributions become negligible when the inte-
grals of (A-2) are evaluated at P = /o, one is left with eq
(12) of [2], with the left side replaced by A/x. Two
wrong signs appearing in eq (12) are corrected in eq
(29a) of [2]. Equation (29a), reproduced here as eq
(A— 4), is used for the general calculation of the en-
hancement ratio, after substituting for /its definition,
eq (1) of this paper. The procedure is one of iteration,
to solve for the mole fraction of air, x (l . This same equa-
tion may be inverted into a quadratic form in B a w>>
to permit evaluation of B mr from experimental enhance-
ment factor data, as was done in [2].

RT In f=g(T,P)+RT In (l-kx„P)+B atl **P

-B wu .(P-e s -xlP)+C a

Xia*

RT

+ C„

Bl

34(1-2*„)P 2

2RT

' y^aww

3xl(l -X„)P 2

3x1 P 2

' ^ innr

RT

(l + 2x„)(l-X;,) 2 P 2 -e?

" D(iat) in,

2RT

*S(i-a* fl )(i-

2RT

x a )P>

B 2

RT

;- (1+3*0(1-

2RT

+ Bmr

+ B inr

Bi

-IxlP-Ba

2xZ(2-3x n )P 2
RT

6x 2 ,(\ -x (l yp 2

RT

2x1(1 -x n ) (I -3x a )P 2
RT

(A-4)

where B im , B ww = the second virial coefficients for air
and water, cm 3 /mol
P,n/=the second interaction virial coeffi-
cient for air-water vapor mixtures,
cm'Vmol
djk = third virial coefficients for air and
water (i=j=k), or third interaction
virial coefficients (cm 3 /mol) 2
R = universal gas constant, 83.1434 bar-
cm : Vmol-K
T= absolute temperature, K
P = total pressure, bar
e s = saturation vapor pressure of the pure

condensed water phase, bar
x ( ,= mole fraction of air in the gas mixture
A: = Henry's "constant," mol fraction/
bar
g(T, P) =the product of the specific volume of
pure phase, at the given (T, P)
condition, and the difference (P —
e,).

one needs, in addition to other equations given in this
paper, the following relationships, discussed in [2]:

72000

^ = 33.97-^^X10 T 1 cm'Vmol

C www = 2.85558 -^ +S^cm 6 /mol 2

C,uru X 10" fi = - 0.20263 + 0.52695 • 10-' 2 ^

-0.74761
+ 0.57576
-0.18065

10-
10

10- 8 * 4 cm 6 /mol 2
t= degrees Celsius, T=t + 273.16

(A-5)

(A-6)

(A-7)

559

Equations (A- 5) and (A-6) are modifications of GofFs
equations [27], which are based on IPTS-48. Use of
r= £ + 273.15 causes insignificant changes in the re-
sults of this paper (see sec. 2).

10. References

[1] Hyland, R. W., and Wexler, A., The enhancement of water

vapor in carbon dioxide-free air at 30, 40, and 50 °C, J. Res.

Nat. Bur. Stand. (U.S.), 77A (Phys. and Chem.), No. 1,

115-131 (Jan.-Feb. 1973).
[2] Hyland, R. W., and Wexler, A., The second interaction (cross)

virial coefficient for moist air, J. Res. Nat. Bur. Stand.

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1973).
[31 Greenspan, L., Low frost-point humidity generator, J. Res.

Nat. Bur. Stand. (U.S.), 77A (Phys. and Chem.), No. 5,

671-677 (Sept.-Oct. 1973).
[4] Russell, H. H., The NBS 1000 lb balance, Report of the 40th

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[5] Wexler, A., and Hyland, R. W., The NBS standard gravimetric

hygrometer (NBS monograph 73, May 1, 1964); also see A.

Wexler and W. A. Wildhack, Eds., Humidity and Moisture,

Vol. Ill (Reinhold Publishing Corp., New York, 1964), p. 389.
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RP1962.
[7] The International Practical Temperature Scale of 1968,

Metrologia 5, 35 (1969).
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(Reinhold Publishing Corp., New York, 1964), p. 8.
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density of ice at °C, J. Res. Nat. Bur. Stand. (U.S.), 38,

583-591 (June 1947) RP1796.
[10] Jakob, M., and Erk, S., Warmeausdehnung des Eises zwischen

und - 253 °C, Wiss. Abh. Physik-Techn. Reichsanst. 12, 301

(1928-29).
[11] Powell, R. W., Preliminary measurements of the thermal

conductivity and expansion of ice, Proc. Roy. Soc. A247,

464 (1958).
[12] Butkovich, T. R., Thermal expansion of ice, J. App. Phys.

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[13] Dantl, Gerhard, Warmeausdehnung von H 2 0- und D 2 0-

Einkristallen, Z. fur Phys. 166, 115 (1962).

[14] LaPlaca, S., and Post, B., Thermal expansion of ice, Acta
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[15] Leadbetter, A. J., The thermodynamic and vibrational properties
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[16] Sengers, J. M. H. L., Klein, M., and Gallagher, J. S. Pressure-
volume-temperature relationships of gases: Virial coefficients,
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[17] Hilsenrath, J., et al., Tables of Thermal Properties of Gases,
Nat. Bur. Stand. (U.S.), Circ. 564 (1955).

[18] Mason, E. A., and Monchick, L., Survey of the equation of
state and transport properties of moist gases, Humidity and
Moisture, Vol. HI, A. Wexler and W. A. Wildhack, Eds.
(Reinhold Publishing Corp., New York, 1965), p. 257.

[19] Loomis, A. G., Solubilities of gases in water, International
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Book Co., New York, 1928), p. 255.

[20] Jones, H. B., Solubilities of various gases in water, Handbook
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[21] Dorsey, N. E., Properties of Ordinary Water Substance (Rein-
hold Publishing Corp., New York, N.Y., 1940).

[22] Wiebe, R., Gaddy, V. L., and Heins, C, Jr., The solubility of
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[23] Krichevsky, I. R., and Kasarnovsky, J. S., Thermodynamical
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[25] Wexler, A., and Greenspan, L., Vapor pressure equation for
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(Paper 79A4-858)

560